Chapter 10 Counting Methods and Probability - 10.2 Use Combinations and the Binomial Theorem - 10.2 Exercises - Skill Practice - Page 694: 18

$2023203$

We know that if we want to select $r$ objects out of $n$ disregarding the order, we can do this in $_nC_r=\frac{n!}{r!(n-r)!}$ ways. Here we can count the number of possibilities by subtracting the number of options when we have $0$ spades from the number of possibilities. Hence here we have $n_1=52,r_1=5$ for all the possibilities and $n_2=39,r_2=5$ for when we have no spades. Hence the answer: $_{52}C_5-{39}C_5=\frac{52!}{47!5!}-\frac{39!}{34!5!}=2598960-575757=2023203$